Dendrite Tip Velocity Research Articles

Prediction of the Secondary Arms Spacing Based on Dendrite Tip Kinetics and Cooling Rate.

Secondary dendrite arm spacing (SDAS) is one of the most important factors affecting macrosegregation and mechanical properties in solidification processes. Predicting SDAS is one of the major parameters in foundry technology. In order to predict the evolution of microstructures during the solidification process, we proposed a simple model which predicted the secondary dendrite arm spacing based solely on the tip velocity (related to the tip supersaturation) and cooling rate. The model consisted of a growing cylinder inside a liquid cylindrical envelope. Two important hypotheses were made: (1) Initially the cylinder radius was assumed to equal the dendrite tip radius and (2) the cylindrical envelope had a fixed radius in the order of the dendrite tip diffusion length. The numerical model was tested against experiments using various Pb-Sn alloys for a fixed temperature gradient. The results were found to be in excellent agreement with experimental measurements in terms of SDAS and dendrite tip velocity prediction. This simple model is naturally destined to be implemented as a sub-grid model in volume-averaging models to predict the local microstructure, which in turn directly controls the mushy zone permeability and macrosegregation phenomena.

Effect of composition on nucleation and growth of equiaxed dendrites in the transparent alloy Neopentylglycol-(d)Camphor during solidification in microgravity conditions

We investigate the effect of composition of hypoeutectic Neopentylglycol-(D)Camphor (NPG-DC) alloy on nucleation and microstructure kinetics during alloy solidification in low gravity conditions without convection and buoyancy effects. Two alloys with compositions of 0.2 and 0.3 wt-frac. DC have been solidified in a small thermal gradient to obtain equiaxed dendritic microstructures with different crystallographic growth orientation. In-situ optical and thermal characterization in the transparent material provided evolution of equiaxed dendrite density, nucleation undercooling, dendrite tip velocity and kinetic law. Significant differences in the results for the two compositions were found, which are discussed in relation to temperature and concentration dependent solid–liquid interfacial properties. For NPG-0.2 wt-frac. DC equiaxed dendrites with 〈100〉 crystallographic growth orientation exist which tend to grow faster and fill the liquid volume quicker compared to the NPG-0.3 wt-frac. DC alloy. This behavior hinders nucleation of further dendrites and decreases their final number. In contrast, for NPG-0.3 wt-frac. DC equiaxed dendrites with 〈111〉 crystallographic growth orientation nucleate at higher undercooling and the kinetic law indicates smaller dendrite tip velocities correlated to a larger number of dendrites.

The boundary integral equation describing a curvilinear solid/liquid interface with nonlinear phase transition temperature

The boundary integral equation (BIE) describes the dynamics of a curved crystallization front separating liquid melt and solid material. We derive a generalized BIE for the thermal‐concentration problem taking into account the nonlinear dependence of the crystallization temperature on solute concentration and the kinetics of atomic attachment at the interface. This equation determines the evolution of the interface function and the equation for crystallization driving force ‐ the melt undercooling at the crystal surface. Our calculations carried out for a dendritic vertex in the form of a paraboloid of revolution have shown that the growth rate of the dendritic tip and its diameter substantially depend on the nonlinear effects under study. In particular, the velocity and diameter of the dendrite tip respectively become greater and narrower with increasing deviation of the liquidus equation from the linear relationship. Also, the dendrite tip velocity can be significantly affected by variations in the exponent of atomic kinetics.

GPU-Based Cellular Automata Model for Multi-Orient Dendrite Growth and the Application on Binary Alloy

To simulate dendrite growth with different orientations more efficiently, a high-performance cellular automata (CA) model based on heterogenous central processing unit (CPU)+ graphics processing unit (GPU) architecture has been proposed in this paper. Firstly, the decentered square algorithm (DCSA) is used to simulate the morphology of dendrite with different orientations. Secondly, parallel algorithms are proposed to take full advantage of many cores by maximizing computational parallelism. Thirdly, in order to further improve the calculation efficiency, the task scheduling scheme using multi-stream is designed to solve the waiting problem among independent tasks, improving task parallelism. Then, the present model was validated by comparing its steady dendrite tip velocity with the Lipton–Glicksman–Kurz (LGK) analytical model, which shows great agreement. Finally, it is applied to simulate the dendrite growth of the binary alloy, which proves that the present model can not only simulate the clear dendrite morphology with different orientations and secondary arms, but also show a good agreement with the in situ experiment. In addition, compared with the traditional CPU model, the speedup of this model is up to 158×, which provides a great acceleration.

Mathematical Modeling of the Solid–Liquid Interface Propagation by the Boundary Integral Method with Nonlinear Liquidus Equation and Atomic Kinetics

In this paper, we derive the boundary integral equation (BIE), a single integrodifferential equation governing the evolutionary behavior of the interface function, paying special attention to the nonlinear liquidus equation and atomic kinetics. As a result, the BIE is found for a thermodiffusion problem of binary melt crystallization with convection. Analyzing this equation coupled with the selection criterion for a stationary dendritic growth in the form of a parabolic cylinder, we show that nonlinear effects stemming from the liquidus equation and atomic kinetics play a decisive role. Namely, the dendrite tip velocity and diameter, respectively, become greater and lower with the increasing deviation of the liquidus equation from a linear form. In addition, the dendrite tip velocity can substantially change with variations in the power exponent of the atomic kinetics. In general, the theory under consideration describes the evolution of a curvilinear crystallization front, as well as the growth of solid phase perturbations and patterns in undercooled binary melts at local equilibrium conditions (for low and moderate Péclet numbers). In addition, our theory, combined with the unsteady selection criterion, determines the non-stationary growth rate of dendritic crystals and the diameter of their vertices.

Competitive growth during directional solidification experiments of 〈1 1 1〉 Dendrites

Competitive growth is a key topic in directional solidification, however, to the authors' best knowledge, there are no experimental or numerical studies on competitive growth between neighbouring 〈111〉 dendrites. The secondary arms of 〈111〉 dendrites can grow with more favourable orientation than those of 〈100〉 dendrites and this has implications for the competitive growth mechanism. An experimental apparatus was developed for in-situ directional solidification with a transparent model alloy, Neopentyl Glycol-35 wt% (D)Camphor (NPG-35wt.%DC), which has 〈111〉 dendrite orientation. Seven tests were conducted with different temperature gradients and isotherm speeds to investigate competitive growth behaviour. Multiple grains were identified in each test with examples of competition at converging and diverging grain boundaries as well as obedience and disobedience with the well-known Walton-Chalmers rule. Unfavourably oriented crystal colonies at diverging grain boundaries were able to outgrow favourably oriented colonies via tertiary branching. Overgrowth by the unfavourably oriented crystal was linked to the advantageous growth directions of secondary arms in 〈111〉 dendrites. Automated dendrite tip velocity tracking further elucidated the tertiary branching mechanism that allowed the unfavourably oriented crystal colonies to stabilise along the growth direction. It is proposed that the deviation from the Walton-Chalmers rule of competitive growth is probabilistic and depends on the initial seeding as well as the 3D nature of the experiments.

A Stable Mode of Dendritic Growth in Cases of Conductive and Convective Heat and Mass Transfer

In this paper, we develop a theory of stable dendritic growth in undercooled melts in the presence of conductive and convective heat and mass transfer boundary conditions at the solid/liquid interface of a dendrite. To simplify the matter and construct the analytical theory, conductive and convective mechanisms are considered separately. Namely, the laws for total undercooling and selection criterion defining the stable growth mode (dendrite tip velocity and diameter) are derived for conductive and convective boundary conditions. To describe the case of simultaneous occurrence of these heat and mass transfer mechanisms, we sew together conductive and convective laws using power stitching functions. The generalised selection theory is compared with experimental data for Al24Ge76 and Ti45Al55 undercooled melts.

Simulation on microstructure evolution of Al−Si alloy under effect of natural convection during solidification

The solidification microstructure of Al−Si alloy was observed in the experiment, the second dendrite arm spacing (SDAS) was measured, and the effect of temperature on the microstructure was analyzed. Phase-field (PF) model incorporating natural convection caused by gravity was employed to simulate the microstructure evolution of Al−Si alloy under the experimental conditions. Good agreements between the experimental and simulation results verified the reliability of the simulation approach proposed in this study. Based on the proposed model, a series of simulation cases (2D and 3D) were performed to investigate the evolution of columnar and equiaxed dendritic structures. It was found that the solute content of the alloy had little impact on the microstructure evolution, while the solute expansion coefficient had obvious effect on the dendrite tip velocities. Significant improvement of computational efficiency was achieved via novel algorithms, making it possible to perform massive simulation for studying the evolution of solidification microstructures, which is hard to be directly observed in experiments via synchrotron radiation for Al−Si alloy.

Mathematical modeling of dendrite growth in an Al–Ge alloy with convective flow

A theory of stable dendrite growth in an undercooled binary melt is developed for the case of intense convection. Conductive heat and mass transfer boundary conditions are replaced by convective conditions, where the flux of heat (or solute) is proportional to the temperature or concentration difference between the surface of the dendrite and far from it. The marginal mode of perturbation wavelengths is calculated using the linear morphological stability analysis. Combining this analysis with the solvability theory, we have derived a selection criterion that represents the first condition to define a combination of dendrite tip velocity and tip diameter. The second condition—the undercooling balance—is derived for intense convection. The theory under consideration determines the dendrite tip velocity and tip diameter for low undercooling. This convective theory is combined with the classical theory of dendritic growth (conductive boundary conditions), which is valid for moderate and high undercooling. Thus, the entire range of melt undercooling is covered. Our results are in good agreement with experiments on Al–Ge crystallization.

Multiple dendrite tip tracking for in-situ directional solidification: Experiments and comparisons to theory

Directional solidification experiments of transparent alloy systems typically show multiple dendrites, a forest of dendrites, growing with preferential alignment. At the length scale of centimetres, an experiment could have hundreds of observable dendrites. Analysis of every dendrite would be laborious and practically difficult to implement. Hence, low numbers of dendrites are routinely selected for analysis as they are assumed to be representative of the growth conditions. Hence, many dendrites go without being analysed. Here, a bespoke experimental apparatus with a novel computer vision algorithm is presented that automatically detects and simultaneously tracks multiple columnar dendrite tips from in-situ video data of directional solidification. The benefits of the algorithm are demonstrated with an application to an experimental test case with the transparent alloy system Neopentyl Glycol-35 wt%D-Camphor (NPG-35 wt%DC). Comparisons of dendrite tip velocity and undercooling measurements with microgravity experimental results from the literature showed notable differences. The current terrestrial data showed similar growth rates but at lower undercoolings (by factors in the range of 0.41–0.68) to that measured in the microgravity experiments. Comparisons were made to the classical Lipton-Glicksman-Kurz (LGK) model and to a modified LGK model adapted with a finite diffusional boundary layer theory to account for convection effects. The modified LGK model showed good agreement for boundary layers between 2.5 and 7.0 µm. An oscillatory component to the tip velocity was observed between adjacent columnar dendrites. Video data of columnar dendritic growth augmented with tip velocity vectors are presented. The tip tracking algorithm is beneficial as, with 385 dendrite tips tracked, it provides statistical and qualitative insights that are otherwise difficult to reconcile using traditional methods.

Phase-field simulations of isomorphous binary alloys subject to isothermal and directional solidification

PurposeIn this work, with a goal to ultimately forward the advancement of additive manufacturing research, the author applies the Wheeler-Boettinger-McFadden model through a progressive series of increasingly complex solidification problems illustrating the evolution of both dendritic as well as columnar growth morphologies. For purposes of convenience, the author assumes idyllic solutions (i.e. the excess energies associated with mixing solid and liquid phases can be neglected).Design/methodology/approachIn this work, the author applied the phase-field model through a progressive series of increasingly complex solidification problems, illustrating the evolution of both dendritic as well as columnar growth morphologies. Beginning with a non-isothermal treatment of pure Ni, the author further examined the isothermal and directional solidification of Cu–Ni binary alloys.Findings(1) Consistent with previous simulation results, solidification simulations from each of the three cases revealed the presence of parabolic, dendrite tips evolving along directions of maximum interface energy. (2) For pure Ni simulations, changes in the anisotropy and noise magnitudes resulted in an increase of secondary dendritic branches and changes in the direction of propagation. The overall shape of the primary structure tended also to elongate with increased anisotropy. (3) For simulations of isothermal solidification of Ni–Cu binary alloys, the development of primary and secondary dendrite arm formation followed similar patterns associated with a pure substance. Calculations of dendrite tip velocity tended to increase monotonically with increasing anisotropy in accordance with previous research. (4) Simulations of directional solidification of Ni–Cu binary alloys with a linear temperature profile demonstrated the presence of cellular dendrites with relatively weak side-branching. The occurrence of solute trapping was also apparent between the primary dendrite columns. Dendrite tip velocities increased with increasing cooling rate.Originality/valueThis research, particularly the section devoted to directional solidification of binary alloys, describes a novel numerical framework and platform for the parametric analysis of various microstructural related quantities, including the effects due to changes in temperature gradient and cooling rate. Both the evolution of the phase and concentration are resolved.

The Interaction between Grains during Columnar-to-Equiaxed Transition in Laser Welding: A Phase-Field Study

A phase-field model was applied to study CET (columnar-to-equiaxed transition) during laser welding of an Al-Cu model alloy. A parametric study was performed to investigate the effects of nucleation undercooling for the equiaxed grains, nucleation density and location of the first nucleation seed ahead of the columnar front on the microstructure of the fusion zone. The numerical results indicated that nucleation undercooling significantly influenced the occurrence and the time of CET. Nucleation density affected the occurrence of CET and the size of equiaxed grains. The dendrite growth behavior was analyzed to reveal the mechanism of the CET. The interactions between different grains were studied. Once the seeds ahead of the columnar dendrites nucleated and grew, the columnar dendrite tip velocity began to fluctuate around a value. It did not decrease until the columnar dendrite got rather close to the equiaxed grains. The undercooling and solute segregation profile evolutions of the columnar dendrite tip with the CET and without the CET had no significant difference before the CET occurred. Mechanical blocking was the major blocking mechanism for the CET. The equiaxed grains formed first were larger than the equiaxed grains formed later due to the decreasing of undercooling. The size of equiaxed grain decreased from fusion line to center line. The numerical results were basically consistent with the experimental results obtained by laser welding of a 2A12 Al-alloy.

Dendritic crystallization from the undercooled melts: effect of tiny amount of impurity

The kinetics of dendrite crystal growth in pure Ni and diluted Ni–B melts is analyzed. Using a model for anisotropic dendrites rapidly growing from undercooled melts, numerical solutions for dendrite tip velocity V and tip radius R are presented. The influence of boron content on tip velocity V and tip radius R of the dendrite growing in Ni-B alloys is shown in comparison with the growth kinetics of pure Ni–dendrites. Particularly, we show that there is a well-known regime of growth in which the impurity (boron) decreases the velocity of the alloying Ni–B–dendrite regarding the nickel dendrite velocity. In addition to this well-established regime, we also show quantitatively that there is a tiny amount of boron which enhances the dendrite velocity in comparison with the velocity of pure nickel dendrite. Explanations of this non-linearity are given using the results of calculations for dendrite tip radii and dendrite growth velocity for alloying Ni–B and pure Ni systems.

On the Theory of Stable Mode of Dendritic Growth in the Presence of Convective Heat and Mass Transfer Boundary Conditions

The dendritic form is one of the most common forms of crystals growing from supercooled melts and supersaturated solutions. In recent decades, an analytical theory has been developed that describes a stable dendrite growth mode under the conditions of a conductive heat and mass transfer process. However, in experiments, the growth of dendritic crystals is often observed under the conditions of convective fluid flow. In the present work, the theory of the growth of dendritic crystals is developed taking into account the convective mechanism of heat and mass transfer at the crystal-melt interface. A stable mode of dendritic growth in the case of intense convective flows near the steady-state growing dendritic tip is analyzed. The selection theory determining a stable growth mode in the vicinity of parabolic solutions as well as the undercooling balance condition are used to find the dendrite tip velocity and its tip diameter as functions of the melt undercooling. It is shown that the theoretical predictions in the case of convective boundary conditions are in agreement with experimental data for small undercoolings. In addition, the convective and conductive heat and mass transfer mechanisms near the growing dendritic surfaces are compared. Our calculations show that the convective boundary conditions essentially influence the stable mode of dendritic growth.

Theoretical modeling of crystalline symmetry order with dendritic morphology

Stable growth of a crystal with dendritic morphology with n-fold symmetry is modeled. Using the linear stability analysis and solvability theory, a selection criterion for thermally and solutally controlled growth of the dendrite is derived. A complete set of non-linear equations consisting of the selection criterion and an undercooling balance (which determines the implicit dependencies of the dendrite tip velocity and tip diameter on the total undercooling) is formulated. The growth kinetics of crystals having different lattice symmetry is analyzed. The model predictions are compared with phase field modeling data on ice dendrites grown from pure undercooled water.

Thermo-solutal growth of a dendritic crystal in the form of an elliptical paraboloid with forced convection

The steady-state analytical solutions describing the thermo-solutal growth of a dendritic crystal in the form of an elliptical paraboloid with allowance for a forced convective flow are found. The system of transcendental equations consisting of the undercooling balance and the stability criterion that describes a stable mode of growth of sections of elliptical dendrites is formulated and analyzed. It is shown that the temperature distributions in the melt, the growth and flow Peclét numbers as well as the dendrite tip diameter and tip velocity in the case of elliptical crystals essentially differ from their analogs in the case of symmetric shapes of dendrites in the form of paraboloids of revolution.

Numerical investigations of six-fold dendritic icing process in subcooled water subject to natural and forced convective environments

Six-fold symmetric ice dendrite growth in subcooled water is investigated based on a GPU-accelerated hybrid lattice Boltzmann (LB) method and the cellular automation (CA) method. A reduction factor is proposed to reduce the discretization induced anisotropy in the hybrid method. The accuracy and feasibility of this hybrid method is validated by comparing simulated tip velocity of ice dendrite with existing experimental data under microgravity conditions, as well as comparing with the LM-K theory under small degrees of subcooling. Effects of natural convection and forced convection environments on dendrite morphology and tip velocity are further studied based on this improved method. It is shown that natural convection promotes ice dendrite to grow upward, and the arm growth velocity can be enhanced or weakened by natural convection, depending on the relative angle between intrinsic growth orientation and gravity direction. Under large natural convection effects, a growth competition phenomenon between different arms on a single ice dendrite is observed which had not been predicted by previous theoretical models. On the other hand, forced convection is shown to have strong enhancing effects on upstream arms and inhibit the growth of downstream arms. The arm growth is always enhanced with increasing inlet Re to various degrees for different relative angles between intrinsic growth orientation and gravity direction.

Phase-field–lattice Boltzmann simulation of dendrite growth under natural convection in multicomponent superalloy solidification

The thermosolutal convection can alter segregation pattern, change dendrite morphology and even cause freckles formation in alloy solidification. In this work, the multiphase-field model was coupled with lattice Boltzmann method to simulate the dendrite growth under melt convection in superalloy solidification. In the isothermal solidification simulations, zero and normal gravitational accelerations were applied to investigate the effects of gravity on the dendrite morphology and the magnitude of melt flow. The solute distribution of each alloy component along with the dendrite tip velocity during solidification was obtained, and the natural convection has been confirmed to affect the microsegregation pattern and the dendrite growth velocity. In the directional solidification simulations, two typical temperature gradients were applied, and the dendrite morphology and fluid velocity in the mushy zone during solidification were analyzed. It is found that the freckles will form when the average fluid velocity in the mushy zone exceeds the withdraw velocity.

Method of evaluation for the non-stationary period of primary dendritic crystallization

Theoretical predictions by many models of solidification kinetics rest on assumption of the steady-state regime of crystal growth with constant velocity. Such approximation holds if the time for solidification of a region under study is much longer than the non-stationarity time for solidification velocity. Even though a lot of experimental evidences which support the existence of long quasi-stationary periods of crystal growth or solidifying layers with constant velocity exists, direct quantitative estimations of such regimes are given usually for planar fronts or essential simplifications which, for instance, do not take into account the finiteness of solidifying bulk. The present work suggests a method to quantify non-stationary periods of crystal growth in comparison with solidification time of finite bulks. In essence we discuss and quantify limits of applicability of steady state crystal growth theories. With this aim, an acceleration- and velocity-dependent interfacial condition [A. Salhoumi, P.K. Galenko, Physica A 447 (2016) 161] is used for the analysis of various regimes of dendrite growth as particular case of crystalline solidification. The time dependence of the dendrite tip velocity is obtained for the growth from a pure (chemically one component) undercooled liquid. The developed theoretical model shows a drastic reducing the non-stationarity time with the increase of undercooling. The present model can be advanced to the arbitrary undercooling and further compared with data of experimental measurements on crystal growth kinetics.

Multiscale dendritic needle network model of alloy solidification with fluid flow

We present a mathematical formulation of a multiscale model for solidification with convective flow in the liquid phase. The model is an extension of the dendritic needle network approach for crystal growth in a binary alloy. We propose a simple numerical implementation based on finite differences and step-wise approximations of parabolic dendritic branches of arbitrary orientation. Results of the two-dimensional model are verified against reference benchmark solutions for steady, unsteady, and buoyant flow, as well as steady-state dendritic growth in the diffusive regime. Simulations of equiaxed growth under forced flow yield dendrite tip velocities within 10% of quantitative phase-field results from the literature. Finally, we perform illustrative simulations of polycrystalline solidification using physical parameters for an aluminum-10wt%copper alloy. Resulting microstructures show notable differences when taking into account natural buoyancy in comparison to a purely diffusive transport regime. The resulting model opens new avenues for computationally and quantitatively investigating the influence of fluid flow and gravity-induced buoyancy upon the selection of dendritic microstructures. Further ongoing developments include an equivalent formulation for directional solidification conditions and the implementation of the model in three dimensions, which is critical for quantitative comparison to experimental measurements.